Effect of thermal stratification on the growth of the internal boundary layer

The growth of a two-dimensional internal boundary layer (IBL), which develops when a neutral or unstably stratified flow over a uniform terrain encounters a step-change in surface roughness, is numerically investigated by a higher-order turbulence closure theory. It is found that the thickness of the IBL increases as ~ x ⁿ, where x is the downstream distance from the roughness-transition line. For a given set of upstream conditions, the value of the exponent n depends only on the Monin-Obukhov length L, and it is approximately independent of the roughness-change parameter M = In (z01/z02). At large fetches, increases markedly with increasing instability.NRC-NAS Resident Research Associate at AFCRL, 1973–74     

The stably stratified internal boundary layer for steady and diurnally varying offshore flow

A two-dimensional numerical mesoscale model is used to investigate the internal structure and growth of the stably stratified internal boundary layer (IBL) beneath warm, continental air flowing over a cooler sea. Two situations are studied — steady-state and diurnally varying offshore flow. In the steady-state case, vertical profiles of mean quantities and eddy diffusion coefficients (K) within the IBL show small, but significant, changes with increasing distance from the coast. The top of the IBL is well defined, with large vertical gradients within the layer and a maximum in the coast-normal wind component near the top. Well away from the coast, turbulence, identified by non-zero K, decreases to insignificant levels near the top of the IBL; the IBL itself is characterised by a critical value of the layer-flux Richardson number equal to 0.18. The overall behaviour of the mean profiles is similar to that found in the horizontally homogeneous stable boundary layer over land.A simple physical model is used to relate the depth of the layer h to several relevant physical parameters viz., x, the distance from the coast and U, the large-scale wind (both normal to the coastline) and g/, being the temperature difference between continental mixed-layer air and sea surface, is the mean potential temperature and g is the acceleration due to gravity. Excellent agreement with the numerical results is found, with h = 0.014x ^1/2 U (g/)^–1/2.In the diurnally varying case, the mean profiles within the IBL show only small differences from the steady-state case, although diurnal variations, particularly in the wind maximum, are evident within a few hundred kilometres of the coast. A mesoscale circulation normal to the coast, and superimposed upon the mean offshore flow, develops seawards of the coastline with maximum vertical velocities about sunset, of depth about 2 km and horizontal scale 500 km. The circulation is related to the advection, and subsequent decay, of daytime convective turbulence over the sea.     

Notes on an Internal Boundary-Layer Height Formula

The derivation of the Panofsky–Dutton internal boundary-layer(IBL) height formula has been revisited. We propose that the upwindroughness length (rather than downwind) should be used in theformula and that a turbulent vertical velocity (_w) ratherthan the surface friction velocity (u_*) should be considered asthe appropriate scaling for the rate of propagation ofdisturbances into the turbulent flow. A published set ofwind-tunnel and atmospheric data for neutral stratification hasbeen used to investigate the influence of the magnitude ofroughness change on the IBL height.     

The internal boundary layer — A review

A review is given of relevant work on the internal boundary layer (IBL) associated with:

1. Small-scale flow in neutral conditions across an abrupt change in surface roughness,
2. Small-scale flow in non-neutral conditions across an abrupt change in surface roughness, temperature or heat/moisture flux,
3. Mesoscale flow, with emphasis on flow across the coastline for both convective and stably stratified conditions.

The major theme in all cases is on the downstream, modified profile form (wind and temperature), and on the growth relations for IBL depth.     

Surface drag and turbulence over an inhomogeneous land surface

Data collected over an inhomogeneous semi-rural area are presented. The data are compared with previous surface-layer data to determine how representative the fixed-point flux measurements are of area averages. Departures from the standard surface-layer results are found to be relatively small (~10–20%), which supports the concept of a blending height above which the flow ceases to respond to variations in the underlying surface and becomes horizontally homogeneous.Effective roughness lengths are derived for different wind directions and the relationship between the effective roughness length and upwind surface is examined in the light of recent ideas on averaging surface roughness lengths. It is found that by averaging drag coefficients, realistic values of the effective roughness length can be calculated which are not very sensitive to the precise choice of the component roughness lengths.     

The structure of the stably stratified internal boundary layer in offshore flow over the sea

Observations obtained mainly from a research aircraft are presented of the mean and turbulent structure of the stably stratified internal boundary layer (IBL) over the sea formed by warm air advection from land to sea. The potential temperature and humidity fields reveal the vertical extent of the IBL, for fetches out to several hundred of kilometres, geostrophic winds of 20–25 m s^–1, and potential temperature differences between undisturbed continental air and the sea surface of 7 to 17 K. The dependence of IBL depth on these external parameters is discussed in the context of the numerical results of Garratt (1987), and some discrepancies are noted.Wind observations show the development of a low-level wind maximum (wind component normal to the coast) and rotation of the wind to smaller cross-isobar flow angles. Potential temperature () profiles within the IBL reveal quite a different structure to that found in the nocturnal boundary layer (NBL) over land. Over the sea, profiles have large positive curvature with vertical gradients increasing monotonically with height; this reflects the dominance of turbulent cooling within the layer. The behaviour is consistent with known behaviour in the NBL over land where curvature becomes negative (vertical gradients of decreasing with height) as radiative cooling becomes dominant.Turbulent properties are discussed in terms of non-dimensional quantities, normalised by the surface friction velocity, as functions of normalised height using the IBL depth. Vertical profiles of these and the normalised wavelength of the spectral maximum agree well with known results for the stable boundary layer over land (Caughey et al., 1979).     

A wind-tunnel study of turbulent flow close to regularly arrayed rough surfaces

Recent observations of flux-gradient anomalies in atmospheric flow close to forests, and similar rough surfaces, prompted a wind-tunnel investigation in which cross-wire anemometry was used to study the vertical development and horizontal variability of adiabatic flow over five regularly arrayed rough surfaces, encompassing a 32-fold range of roughness concentration . The roughness elements were cylinders, 6 mm in both height and diameter.Below a layer in which the velocity profile is semi-logarithmic, two surface influences upon the mean velocity field can be distinguished: wake diffusion and horizontal inhomogeneity. The wake diffusion effect causes non-dimensional vertical velocity gradients to be smaller than in the semi-logarithmic region; at least for elements with aspect ratios l/h 1, it is governed by the transverse dimension l of the roughness elements, and is observed when z > h + 1.5l (where z is height above the underlying surface, and h is the height of the roughness elements). A simple diffusivity model successfully describes the horizontally averaged velocity profiles in the region of wake influence, despite conceptual disadvantages. The horizontal inhomogeneity of the flow is negligible when z > h + D (D being the inter-element spacing), and does not entirely mask the wake diffusion effect except over very sparsely roughened surfaces ( 0.02). A criterion for negligibility of both effects, and hence for applicability of conventional turbulent diffusivity theory for momentum, is z > h + 1.5D. These results are compared with atmospheric data, and indicate that wake diffusion may well cause some underestimation of the zero-plane displacement d over typical vegetated surfaces.     

Markov chain simulations of vertical dispersion in the neutral surface layer for surface and elevated releases

Vertical dispersion in the neutral surface layer is investigated using a Markov Chain simulation procedure. The conceptual basis of the procedure is discussed and computation procedures outlined. Wind and turbulence parameterizations appropriate to the neutral surface layer are considered with emphasis on the Lagrangian time scale. Computations for a surface release are compared with field data. Good agreement is found for the variation of surface concentration and cloud height to distances 500 m downwind of the source. The functional form of the vertical concentration profile is examined and an exponential with exponent 1.6 is found to give the best fit with simulations.For elevated releases, it is demonstrated that an initial dip of the mass mean height from the simulation can be normalized for various release heights using a non-dimensionalized downwind coordinate incorporating advective wind speed and wind shear. The vertical distribution standard deviation ( _z ), as employed in Gaussian models, shows a fair degree of independence with source height but close examination reveals an optimum source height for maximum _z at a given downwind distance,x. This source height increases with downwind distance. Also the simulations indicate that vertical wind shear is more important than vertical variation of Lagrangian time scale close to the source, with a reverse effect farther downwind.     

Footprint prediction of scalar fluxes from analytical solutions of the diffusion equation

The use of analytical solutions of the diffusion equation for footprint prediction is explored. Quantitative information about the footprint, i.e., the upwind area most likely to affect a downwind flux measurement at a given height z, is essential when flux measurements from different platforms, particularly airborne ones, are compared. Analytical predictions are evaluated against numerical Lagrangian trajectory simulations which are detailed in a companion paper (Leclerc and Thurtell, 1990). For neutral stability, the structurally simple solutions proposed by Gash (1986) are shown to be capable of satisfactory approximation to numerical simulations over a wide range of heights, zero displacements and roughness lengths. Until more sophisticated practical solutions become available, it is suggested that apparent limitations in the validity of some assumptions underlying the Gash solutions for the case of very large surface roughness (forests) and tentative application of the solutions to cases of small thermal instability be dealt with by semi-empirical adjustment of the ratio of horizontal wind to friction velocity. An upper limit of validity of these solutions for z has yet to be established.     

Wind profile development above a locally adjusted sea surface

Results are presented from a numerical experiment of wind and shear stress profile development away from a shore line; the water surface is assumed to obey the Charnock-Ellison relation between surface roughness and friction velocity. In typical cases the upwind, land surface is rough relative to the sea and the velocity and shear stress results are qualitatively similar to those for flows from relatively rough to relatively smooth solid surfaces. In the present case, however, the downwind surface roughness and friction velocity vary with position and we find that wind profile development may play a significant role in the relationship between sea surface roughness and fetch.     

Roughness effects on urban turbulence parameters

Urban roughness lengths are estimated from measurements of u and u _* at one level under neutral conditions, assuming a logarithmic form for the vertical profile of wind velocity. At a given location in the urban area, estimated values show considerable directional variation. The dependence of some turbulence parameters on the urban roughness lengths is experimentally investigated during near-neutral conditions. The ratios _i /u _* decrease with roughness whereas the turbulence intensities _i /u increase with it. The dependence on roughness is not the same for all components.     

A scheme for computing surface fluxes from mean flow observations

A computational scheme is developed for estimating turbulent surface stress, sensible heat flux and humidity flux from mean velocity, temperature and humidity at a single height in the atmospheric surface layer; conditions at this reference level are presumed known from observations or from a numerical atmospheric circulation model. The method is based on coupling a Monin-Obukhov similarity profile to a force-restore formulation for the evolution of surface soil temperature to yield the local values of shear stress, heat flux and surface temperature. A self-contained formulation is presented including parameterizations for solar and infrared radiant flux at the surface.In addition to reference-level mean flow properties, the parameters needed to implement the scheme are thermal heat capacity of the soil, surface aerodynamic roughness, latitude, solar declination, surface albedo, surface emissivity and atmospheric transmissivity.Sample calculations are presented for (a), constant atmospheric forcing at the reference level, and (b) variable atmospheric forcing corresponding to Kahle's (1977) measurements of windspeed, air temperature and radiometer soil surface temperature under dry vegetatively sparse conditions in the Mohave Desert in California. The latter case simulated the observed diurnal variations resonably well for the parameters used.Consultant, Atmospheric Sciences Division, Department of Energy and Environment, Brookhaven National Laboratory, Upton, N.Y., pc11973, U.S.A.     

Turbulence Characteristics Over Complex Terrain In West China

Meteorological data of velocity components and temperature have been measured on a mast of height 4.9 m at one site in the Heihe River Basin Field Experiment (HEIFE) conducted in west China. Mean and individual turbulence parameters, power spectra/cospectra, phase angles and their changes withfetch downwind of a change in surface roughness were analyzed. The turbulence characteristics depend strongly on the prevailing wind direction, which in turn is associated with changes in the upwind surface roughness pattern. The results show that values of horizontal velocity standard deviations sigma;_u,v scaled with local friction velocity u under different stratifications are larger than those over flat terrain, while the values of _w/u have the same values as over flat terrain. The differences between variance values of the horizontal velocity components, u and v, over inhomogeneous terrain were found to be significantly smaller than those over flat terrain. Since energy densities of the w spectra, uw and wT cospectra at low frequencies are relatively lower than those of longitudinal velocity spectra, total energies of w spectra, uw and wT cospectra tend to be in equilibrium with the local terrain. The values of phase angles at the low frequency end of the frequency showed obvious differences associated with changes of roughness.     

Turbulence characteristics along several towers

Observations from several towers are used to show how measurements of wind and temperature near the surface can be used to estimate the variances of the horizontal velocity and the dissipation rate up to the top of the towers, provided the roughness length is known. The roughness length usually varies with wind direction, and the traditional methods of estimating it tend to lead to over-estimates.Analysis of cross spectra between velocity components at different levels shows that Davenport's Geometrical Similarity is satisfied. Coherence falls off exponentially with the ratio of height interval to wavelength, and the decay parameter depends on Richardson number near the surface. Coherences at different sites show no significant differences in neutral air. The lateral velocity components have larger coherence and more time delay between levels than the horizontal components at all sites.Time delay and coherence are also discussed in other Cartesian directions, and it is suggested that these quantities, having relatively simple properties, can be used as building blocks for an empirical three-dimensional model of turbulence.Contribution No. 6951     

Diabatic wind speed profiles in coastal regions: Comparison of an internal boundary layer (IBL) model with observations

A model is presented to transform wind speed observations at a single height over sea or near the coast to any possible location and height in a topographic flat coastal region (up to distances of about 5 km from the coast and up to heights of 100 m). Only moderate and strong winds from the sea are considered, which are particularly important for wind energy applications. The model, called diabatic coast model, which is based on the well known internal boundary layer (IBL) concept and Monin-Obukhov similarity theory, describes the effects of the roughness transition from sea to land as well as the effect of stability on the shape of the profiles and the IBL growth. The predicted IBL heights are compared with published data.In the second part of this paper, the model is compared with measurements taken at the Maasvlakte location near the Dutch coast. It is shown that a neutral formulation of the IBL height is sufficient to model the overall mean wind speed with height, but that stability corrections are needed to describe the diurnal variations in wind speed properly. Finally, an application is given, where a single routine wind speed observation at the coast, combined with air-water temperature differences is used to predict the wind speed at 500m from the coast at heights of 10 and 53 m. The results are in good agreement with the measurements.     

Numerical simulation of turbulent convective flow over wavy terrain

By means of a large-eddy simulation, the convective boundary layer is investigated for flows over wavy terrain. The lower surface varies sinusoidally in the downstream direction while remaining constant in the other. Several cases are considered with amplitude up to 0.15H and wavelength ofH to 8H, whereH is the mean fluid-layer height. At the lower surface, the vertical heat flux is prescribed to be constant and the momentum flux is determined locally from the Monin-Obukhov relationship with a roughness lengthz _o=10^–4 H. The mean wind is varied between zero and 5w _*, wherew _* is the convective velocity scale. After rather long times, the flow structure shows horizontal scales up to 4H, with a pattern similar to that over flat surfaces at corresponding shear friction. Weak mean wind destroys regular spatial structures induced by the surface undulation at zero mean wind. The surface heating suppresses mean-flow recirculation-regions even for steep surface waves. Short surface waves cause strong drag due to hydrostatic and dynamic pressure forces in addition to frictional drag. The pressure drag increases slowly with the mean velocity, and strongly with /H. The turbulence variances increase mainly in the lower half of the mixed layer forU/w _*>2.     

Interaction between the atmospheric and oceanic boundary layers

The two-layer system of an atmosphere over water bodies is reduced to a single-layer problem. Values of the interfacial quantities, such as the friction velocity, the surface velocity, the angles, and , between the surface shear stress and the geostrophic wind velocity and the surface wind velocity, respectively, and the surface roughness, all of which depend upon external parameters, such as the geostrophic wind and stratifications, are obtained. The geostrophic drag coefficient C _d, the geostrophic wind coefficient C _f, and the angles , and , of the turbulent flow at the sea-air interface are functions of a dimensionless number, mfG/kg, with S ₁ and S ₂ as two free stratification parameters. The surface roughness is uniquely determined from the geostrophic wind rather than from the wind profile in the boundary layer.Formerly Visiting Research Associate, Applied Physics Branch, Earth Observations Division, NASA-Manned Spacecraft Center, Houston, Texas.     

下垫面特性突变时的内边界层数值研究

本文建立了一个考虑层结的二维内边界层数值模式,研究了下垫面粗糙度、温度、湿度突变引起的内边界层。结果表明:下垫面温度突变引起的温度内边界层与下垫面粗糙度突变引起的速度内边界层具有明显的差异。     

Numerical simulation of internal boundary-layer development and comparison with atmospheric data

A finite-volume numerical model is employed to investigate the adaptation of the atmospheric boundary layer to a change in the underlying surface roughness, such as that existing in the transition from land to the free surface of a water body. Numerical results are validated by comparison with neutral stratification atmospheric data and compared with the internal boundary-layer (IBL) heights computed using a number of existing empirical formulae. The numerical analysis allows an extension of the fetch range in which the existing formulae, calibrated only by comparison with short fetch data, may be applied. An argument is offered that the spatial variability of the water surface roughness should be also taken into account for the IBL development over the surface of a water body.     

A new theory of the energy spectrum

In a recent paper, the author introduced a new viscous boundary layer, called the mesolayer, in turbulent shear flow. Its importance stems from its location between the inner and outer regions which are controlled by the law of the wall and Reynolds number similarity, respectively. This intrusion prevents the classical overlap assumption which appears to be fundamental in the derivation of the classical logarithmic behavior. The mesolayer has a thickness proportional to Taylor's microscale . This, and the analogy between the energy equation for the spectrum function of isotropic turbulence and the momentum equation for shear flow, suggest the existence of a similar region in wavenumber space with wavenumber k ~ ^-1. This mesoregion separates the inner region k ~ k _s(where k _s^-1 and is the Kolmogorov length) and the outer region k k _e(where k _e ^-1 and l is the energy-containing eddy size) and again invalidates the overlap assumption which appears to be fundamental in the derivation of the classical k ^-5/3-behavior of the inertial subrange.Incorporation of the mesoregion into the argument leads to a new theory with k ^-5/3-behavior in two regions (^-1 k k _s) and (k _e k ^-1) although with two different coefficients of proportionality (Kolmogorov constants). This leads to a wandering of the spectrum curve about the classical k ^-5/3 line similar to a wandering in turbulent shear flow about the logarithmic curve. This is clearly indicated by the data for the variation of the Kolmogorov constant.Other data support the new theory. In particular, the location of the point k _mwhere the curve of the nonlinear energy-transfer function goes through zero shows agreement with the theory, i.e., k _m^-1.